Definition
Given a set of k + 1 data points
where no two are the same, the interpolation polynomial in the Lagrange form is a linear combination
of Lagrange basis polynomials
Note how, given the initial assumption that no two are the same, so this expression is always well-defined. The reason pairs with are not allowed is that no interpolation function such that would exist; a function can only get one value for each argument . On the other hand, if also, then those two points would actually be one single point.
For all, includes the term in the numerator, so the whole product will be zero at :
On the other hand,
In other words, all basis polynomials are zero at, except, because it lacks the term.
It follows that, so at each point, showing that interpolates the function exactly.
Read more about this topic: Lagrange Polynomial
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)